So, as I understand your proposal, one does not even need to be a miner to implement this.
Correct.
Shareholders don't need any robot code:
1) ASICMINER runs the "shareholder registry" robot.
2) Shareholders do not need anything but a bitcoin client with the "sign message" feature.
3) A person-to-person share trade looks like this:
a) Alice sends 100 BTC to Bob
b) Alice sends bitcoin address ABCDEF to Bob
c) Bob emails ASICMINER robot, with signed message "transfer ownership of my 50 shares to address ABCDEF"
d) The next time ASICMINER robot publishes a shareholder registry, it says "50 shares owned by ABCDEF"
4) If you want to run an automated share trading market, create a website that accepts bitcoin or ASICMINER share deposits, ....